tailieunhanh - báo cáo hóa học:" Research Article Approximate Controllability of Abstract Discrete-Time Systems"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Approximate Controllability of Abstract Discrete-Time Systems | Hindawi Publishing Corporation Advances in Difference Equations Volume 2010 Article ID 695290 17 pages doi 2010 695290 Research Article Approximate Controllability of Abstract Discrete-Time Systems Hernan R. Henriquez1 and Claudio Cuevas2 1 Departamento de Matematica Universidad de Santiago USACH Casilla 307 Correo 2 Santiago Chile 2 Departamento de Matemdtica Universidade Federal de Pernambuco Recife PE 50540-740 Brazil Correspondence should be addressed to Claudio Cuevas cch@ Received 12 April 2010 Accepted 2 September 2010 Academic Editor Rigoberto Medina Copyright 2010 H. R. Henriquez and C. Cuevas. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. Approximate controllability for semilinear abstract discrete-time systems is considered. Specifically we consider the semilinear discrete-time system Xk 1 AkXk f k Xk BkUk k e N0 where Ak are bounded linear operators acting on a Hilbert space X Bk are X-valued bounded linear operators defined on a Hilbert space U and f is a nonlinear function. Assuming appropriate conditions we will show that the approximate controllability of the associated linear system Xk 1 AkXk BkUk implies the approximate controllability of the semilinear system. 1. Introduction In this paper we deal with the controllability problem for semilinear distributed discrete-time control systems. In order to specify the class of systems to be considered we set X for the state space and U for the control space. We assume that X and U are Hilbert spaces. Moreover throughout this paper we denote by Ak X X bounded linear operators Bk U X k e No bounded linear maps that represent the control action and f N0 X X X a map such that f k is continuous for each k e N0. Furthermore Ak Bk and f satisfy appropriate conditions which will be specified later. We will study the .